On the conditional joint probability distributions of phase-type under the mixture of finite-state absorbing Markov jump processes

Abstract: This paper presents some new results on the conditional joint probability distributions of phase-type under the mixture of right-continuous Markov jump processes with absorption on the same finite state space $\mathbb{S}$ moving at different speeds, where the mixture occurs at a random time. Such mixture was first proposed by Frydman \cite{Frydman2005} and Frydman and Schuermann \cite{Frydman2008} as a generalization of the mover-stayer model of Blumen et at. \cite{Blumen}, and was recently extended by Surya \cite{Surya2018}. When conditioning on all previous and current information $\mathcal{F}{t,i}=\mathcal{F}{t-}\cup{X_t=i}$, with $\mathcal{F}{t-}={X_s, 0<s\leq t-}$ and $i\in\mathbb{S}$, of the mixture process $X$, distributional identities are explicit in terms of the Bayesian updates of switching probability, the likelihoods of observing the sample paths, and the intensity matrices of the underlying Markov processes, despite the fact that the mixture itself is non-Markov. They form non-stationary function of time and have the ability to capture heterogeneity and path dependence. When the underlying processes move at the same speed, in which case the mixture reduces to a simple Markov jump process, these features are removed, and the distributions coincide with that of given by Neuts \cite{Neuts1975} and Assaf et al. \cite{Assaf1984}. Furthermore, when conditioning on $\mathcal{F}{t-}$ and no exit to the absorbing set has been observed at time $t$, the distributions are given explicitly in terms of an additional Bayesian updates of probability distribution of $X$ on $\mathbb{S}$. Examples are given to illustrate the main results.